Abstract Spence [9] constructed $$\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) $$ 3 d + 1 ( 3 d + 1 - 1 ) 2 , 3 d ( 3 d + 1 + 1 ) 2 , 3 d ( 3 d + 1 ) 2 -difference sets in groups $$K \times C_3^{d+1}$$ K × C 3 d + 1 for d any positive integer and K any group of order $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 . Smith and Webster [8] have exhaustively studied the $$d=1$$ d = 1 case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in $$A_4 \times C_3$$ A 4 × C 3 by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to $$C_3^2$$ C 3 2 . Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to $$C_3^{d+1}$$ C 3 d + 1 as long as $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We conjecture that any group of order $$\frac{3^{d+1}(3^{d+1}-1)}{2}$$

Complex orthogonal designs (CODs) of rate 1/2 have been considered recently for use in analog transmissions and as an alternative to maximum rate CODs due to the savings in decoding delay as the number of antennas increases. While algorithms have been developed to show that an upper bound on the minimum decoding delay for rate 1/2 CODs withn=2m-1 orn=2mcolumns is ν(n) = 2m-1 or ν(n) = 2m, depending on the parity ofnmodulo 8, it remains open to determine the exact minimum delay. This paper shows that this bound ν(n) is also a lower bound on minimum decoding delay for a major class of rate 1/2 CODs, named balanced complex orthogonal designs (BCODs), and that this is the exact minimum decoding delay for most BCODs. These rate 1/2 codes are conjugation-separated and thus permit a linearized description of the transceiver signal. BCODs also display other combinatorial properties that are expected to be useful in implementation, such as having no linear processing. An elegant construction is provided for a class of rate 1/2 CODs that have no zero entries, effectively no irrational coefficients, no linear processing, and have each variable appearing exactly twice per column. The resulting codes meet the aforementioned bound on decoding delay in most cases. This class of CODs will be useful in practice due to their low peak-to-average power ratio (PAPR) and other desirable properties.

Due to the low peak-to-mean envelope power ratio (PMEPR) of Golay sequences together with their connection to generalized Reed-Muller codes, many authors have proposed using Golay sequences as codewords in orthogonal frequency-division multiplexing (OFDM). Golay sequences that unexpectedly share the same aperiodic autocorrelation function can be used to construct longer Golay sequences. We study the shared autocorrelation property of general sequences with the ultimate goal of using those results to find other low power sequences. In this paper, we provide several new constructions of families of pairs of sequences with the shared autocorrelation property that enable us to explain nearly all of the binary shared autocorrelation property for sequences of lengths up to 24.

In 1995, the IEEE approved the 802.12 standard for data transmission at 100-Mbit/s using the Demand Priority Network Access protocol. 100 VG-AnyLAN products conforming to this standard offered an upgrade path for Ethernet and token ring networks, without requiring new building wiring. A key factor in the approval of the 802.12 standard was the demonstrated error detection properties of its coding scheme. In particular, the coding scheme allows the detection of error bursts affecting encoded data carried on four parallel conductors, using nothing more than the standard IEEE 32-bit cyclic redundancy check applied to the unencoded data. Although these error detection properties were presented for verification as part of the standards process, for many years commercial considerations prevented public disclosure of how the code was actually found. These considerations no longer apply, and, in this paper, we explain in detail the design principles of the code, combining geometrical insight, linear algebra, combinatorial reasoning, and computer search.

We present a range of coding schemes for OFDM transmission using binary, quaternary, octary, and higher order modulation that give high code rates for moderate numbers of carriers. These schemes have tightly bounded peak-to-mean envelope power ratio (PMEPR) and simultaneously have good error correction capability. The key theoretical result is a previously unrecognized connection between Golay complementary sequences and second-order Reed-Muller codes over alphabets Z/sub 2/h. We obtain additional flexibility in trading off code rate, PMEPR, and error correction capability by partitioning the second-order Reed-Muller code into cosets such that codewords with large values of PMEPR are isolated. For all the proposed schemes we show that encoding is straightforward and give an efficient decoding algorithm involving multiple fast Hadamard transforms. Since the coding schemes are all based on the same formal generator matrix we can deal adaptively with varying channel constraints and evolving system requirements.

This letter describes a technique for finding cyclic redundancy check polynomials for systems for transmission over symmetric channels which encode information in multiple voltage levels, so that the resulting redundancy check gives good error protection and is efficient to implement. The codes which we construct have a Hamming distance of 3 or 4. We discuss a way to reduce burst error in parallel transmissions and some tricks for efficient implementation of the shift register for these polynomials. We illustrate our techniques by discussing a particular example where the number of levels is 9, but they are applicable in general.